Skip to main content

Improved Uniform Error Bounds on a Lawson-type Exponential Integrator Method for Long-Time Dynamics of the Nonlinear Double Sine-Gordon Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A Lawson-type exponential integrator combined with the Fourier pseudo-spectral method is provided for the nonlinear Double Sine-Gordon equation (DSGE), while the nonlinearity is characterized by \(\beta /\epsilon \) with small parameter \(\epsilon \in (0,1]\) and interaction parameter \(\beta \in (0,+\infty )\). In comparison to the Sine-Gordon equation, DSGE has many properties of solitons as well as its own unique new features. This is the first work to numerically simulate the physical phenomena arising from DSG kinks collisions. The improved uniform error bounds are proved by using the regularity compensation oscillatory (RCO) technique, which are \(O(\alpha ^2\tau +h^m)\) up to the long time at \(T_{\epsilon }=T/\alpha ^2\), where \(\alpha =\epsilon \) for \(\beta \ge 1\) and \(\alpha ={\epsilon }/{\beta }\) for \(0<\epsilon<\beta <1\). Based on the uniform bounds, the error estimation for the discrete energy is derived. Furthermore, the improved error bounds are extended to two oscillatory DSGEs with \(O(\epsilon ^2)\) and \(O(\epsilon ^2/\beta ^2)\) wavelength in time. Numerical examples are provided to illustrate the accuracy and discrete energy property of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data Availibility

Data will be made available on request.

References

  1. Aktosun, T., Demontis, F., Cornelis, V.D.M.: Exact solutions to the Sine-Gordon equation. J. Math. Phys. 51(12), 1262 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atanasova, P.K., Boyadjiev, T., Shukrinov, Y.M., Zemlyanaya, E.: Numerical modeling of long josephson junctions in the frame of double Sine-Gordon equation. Math. Models Comput. Simul. 3(3), 389–398 (2011)

    Article  MATH  Google Scholar 

  3. Bao, W., Feng, Y., Yi, W.: Long time error analysis of finite difference time domain methods for the nonlinear Klein-Gordon equation with weak nonlinearity. Commun. Comput. Phys. 26(5), 1307–1334 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bao, W., Cai, Y., Feng, Y.: Improved uniform error bounds on time-splitting methods for long-time dynamics of the nonlinear Klein-Gordon equation with weak nonlinearity. SIAM J. Numer. Anal. 60(4), 1962–1984 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bao, W., Cai, Y., Feng, Y.: Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schrödinger equation. Math. Comput. 92(341), 1109–1139 (2023)

    Article  MATH  Google Scholar 

  6. Bin, H., Qing, M., Yao, L., Weiguo, R.: New exact solutions of the double Sine-Gordon equation using symbolic computations. Appl. Math. Comput. 186(2), 1334–1346 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Bullough, R., Caudrey, P., Gibbs, H.: The double Sine-Gordon equations: A physically applicable system of equations. In: Solitons, vol 17, Springer, pp 107–141 (1980)

  8. Dikandé, A., Kofané, T.: Double Sine-Gordon solitons in two-dimensional crystals. J. Phys. Condens Mat. 7(10), L141 (1995)

    Article  MATH  Google Scholar 

  9. Domairry, G., Davodi, A.G., Davodi, A.G.: Solutions for the double Sine-Gordon equation by exp-function, tanh, and extended tanh methods. Numer Methods Partial Differential Equations 26(2), 384–398 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Duckworth, S.: Radiative Processes in Multi-Level and Degenerate Atomic Systems. The University of Manchester (United Kingdom) (1976)

  11. Duckworth, S., Bullough, R.K., Caudrey, P.J., Gibbon, J.D.: Unusual soliton behaviour in the self-induced transparency of q(2) vibration-rotation transitions. Phys. Lett. A 57(1), 19–22 (1976)

    Article  MATH  Google Scholar 

  12. Fang, D., Zhang, Q.: Long-time existence for semi-linear Klein-Gordon equations on tori. J. Differential Equations 249(1), 151–179 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Faou, E., Schratz, K.: Asymptotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime. Numer. Math. 126(3), 441–469 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Feng, Y.: Long time error analysis of the fourth-order compact finite difference methods for the nonlinear Klein-Gordon equation with weak nonlinearity. Numer Methods Partial Differential Equations 37(1), 897–914 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Feng, Y., Schratz, K.: Improved uniform error bounds on a lawson-type exponential integrator for the long-time dynamics of Sine-Gordon equation. (2022) DOI: https://doi.org/1048550/arXiv:221109402

  16. Feng, Y., Yi, W.: Uniform error bounds of an exponential wave integrator for the long-time dynamics of the nonlinear Klein-Gordon equation. Multiscale Model. Simul. 19(3), 1212–1235 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Feng, Y., Maierhofer, G., Schratz, K.: Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations. Math. Comput. 93(348), 1569–1598 (2024)

    Article  MATH  Google Scholar 

  18. Gani, V.A., Marjaneh, A.M., Askari, A., Belendryasova, E., Saadatmand, D.: Scattering of the double Sine-Gordon kinks. Eur. Phys. J. C 78, 345 (2018)

    Article  MATH  Google Scholar 

  19. Hong, B.: New jacobi elliptic function solutions for the double Sine-Gordon equation. World J. Model 7(2), 133–138 (2009)

    MATH  Google Scholar 

  20. Zagrodziński, J.: Particular solutions of the Sine-Gordon equation in 2 + 1 dimensions. Phys. Lett. A 72(4–5), 284–286 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  21. Szeftel, J.: Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres. Int. Math. Res. Not IMRN 2004(37), 1897–1966 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kasai, Y., Tanda, S., Hatakenaka, N., Takayanagi, H.: Fluxon dynamics in isolated long josephson junctions. Physica C 352(1–4), 211–214 (2001)

    Article  MATH  Google Scholar 

  23. Kitchenside, P.W.: The double Sine-Gordon equation: Numerical and perturbative studies. PhD thesis, University of Manchester Institute of Science and Technology (UMIST) (1979)

  24. Leibbrandt, G.: New exact solutions of the classical Sine-Gordon equation in 2 + 1 and 3 + 1 dimensions. Phys. Rev. Lett. 41(7), 435–438 (1978)

    Article  MATH  Google Scholar 

  25. Li, J., Zhu, L.: A uniformly accurate exponential wave integrator fourier pseudo-spectral method with structure-preservation for long-time dynamics of the Dirac equation with small potentials. Numer Algorithms 92(2), 1367–1401 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mussardo, G., Riva, V., Sotkov, G.: Semiclassical particle spectrum of double Sine-Gordon model. Nuclear Phys. B 687(3), 189–219 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nikolay, K.: Breather and soliton wave families for the Sine-Gordon equation. Proc. Rco. Lond A. 454(1977), 2409–2423 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Peng, Y.Z.: Exact solutions for some nonlinear partial differential equations. Phys. Lett. A 314(5–6), 401–408 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Peyravi, M., Montakhab, A., Riazi, N., Gharaati, A.: Interaction properties of the periodic and step-like solutions of the double Sine-Gordon equation. Eur. Phys. J. B 72, 269–277 (2009)

    Article  MATH  Google Scholar 

  30. Popov, S.P.: Influence of dislocations on kink solutions of the double Sine-Gordon equation. Comput. Math. Math. Phys. 53(12), 1891–1899 (2013)

    Article  MATH  Google Scholar 

  31. Takács, G., Wágner, F.: Double Sine-Gordon model revisited. Nuclear Phys. B 741(3), 353–367 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, H., Fu, Y.: Exact traveling wave solutions for the (2+1)-dimensional double Sine-Gordon equation using direct integral method. Appl. Math. Lett. 146(108), 798 (2023)

    MathSciNet  MATH  Google Scholar 

  33. Wang, M., Li, X.: Exact solutions to the double Sine-Gordon equation. Chaos, Solitons Fractals 27(2), 477–486 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, Y., Zhao, X.: A symmetric low-regularity integrator for nonlinear Klein-Gordon equation. Math. Comput. 91(337), 2215–2245 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wazwaz, A.M.: The tanh method: exact solutions of the Sine-Gordon and the sinh-gordon equations. Appl. Math. Comput. 167(2), 1196–1210 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wazwaz, A.M.: The tanh method and a variable separated ode method for solving double Sine-Gordon equation. Phys. Lett. A 350(5–6), 367–370 (2006)

    Article  MATH  Google Scholar 

  37. Wu, Y., Yao, F.: A first-order fourier integrator for the nonlinear Schrödinger equation on \(\mathbb{T} \) without loss of regularity. Math. Comput. 91(335), 1213–1235 (2022)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their appreciation to the anonymous reviewer for the invaluable comments that have greatly improved the quality of the manuscript. This work was supported by Guangdong Basic and Applied Basic Research Foundation, China (Grant No. 2024A1515012548) and the National Natural Science Foundation of China (Nos. 12471374, 12171148).

Author information

Authors and Affiliations

Authors

Contributions

Ling Zhang: Formal analysis, Data Curation, Software, Validation, Writing-Original Draft. Huailing Song: Conceptualization, Methodology, Writing Reviewing & Editing, Project Administration, Supervision. Wenfan Yi: Formal analysis, Validation, Writing Reviewing & Editing

Corresponding authors

Correspondence to Huailing Song or Wenfan Yi.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

The proof of Lemma 1

By the definition of the function f and g, we have

$$\begin{aligned} F(\psi )=i{\langle \partial _x \rangle }^{-1} g(\psi ) =\mathscr {H}(\psi )+i{\langle \partial _x \rangle }^{-1}\epsilon ^4 r_1(\psi )+i{\langle \partial _x \rangle }^{-1}\dfrac{\epsilon ^4}{\beta ^4}r_2(\psi ). \end{aligned}$$
(1)

According to (9)and (10), we can write the local truncation error as

$$\begin{aligned} {\mathscr {E}}^n&=\tau e^{i\tau \langle \partial _x \rangle } F\big (\psi (t_n)\big )-\int _0^{\tau }e^{i(\tau -\sigma )\langle \partial _x\rangle }F\big (\psi (t_n+\sigma )\big )d\sigma \\&=\tau e^{i\tau \langle \partial _x \rangle } \Big ({\mathscr {H}}\big (\psi (t_n)\big )+i\epsilon ^4 {\langle \partial _x \rangle }^{-1}r_1\big (\psi (t_n)\big )+i\dfrac{\epsilon ^4}{\beta ^4}{\langle \partial _x \rangle }^{-1}r_2\big (\psi (t_n)\big )\Big )-\int _0^{\tau } e^{i(\tau -\sigma )\langle \partial _x\rangle }\\&\quad \Big ({\mathscr {H}}\big (\psi (t_n+\sigma )\big )+i\epsilon ^4 {\langle \partial _x \rangle }^{-1}r_1\big (\psi (t_n+\sigma )\big )+i\dfrac{\epsilon ^4 }{\beta ^4}{\langle \partial _x \rangle }^{-1}r_2\big (\psi (t_n+\sigma )\big )\Big )d\sigma \\&= e^{i\tau \langle \partial _x \rangle } \Big ( \tau {\mathscr {H}}\big (\psi (t_n)\big )-\int _0^{\tau }e^{-i\sigma \langle \partial _x \rangle }{\mathscr {H}}\big (e^{i\sigma \langle \partial _x\rangle }\psi (t_n)\big )d\sigma \Big )+K_1+K_2+K_3\\&={\mathscr {Q}}(\psi (t_n))+{\mathscr {R}}^n, \end{aligned}$$

where \(\mathscr {Q}\) is defined in (22), \(\mathscr {R}^n=K_1+K_2+K_3\), and the corresponding representations are as follows

$$\begin{aligned} K_1&=i\epsilon ^4\tau e^{i\tau \langle \partial _x \rangle }{\langle \partial _x\rangle }^{-1} r_1\big (\psi (t_n)\big )+i\dfrac{\epsilon ^4}{\beta ^4}\tau e^{i\tau \langle \partial _x \rangle }{\langle \partial _x\rangle }^{-1} r_2\big (\psi (t_n)\big ),\\ K_2&=-\int _0^{\tau } e^{i(\tau -\sigma )\langle \partial _x\rangle }\int _0^{1}(1-\xi )H'\Big (e^{i\tau \langle \partial _x\rangle }\psi (t_n)\\&\quad +\xi \int _0^{\sigma }e^{i(\sigma -\theta )\langle \partial _x\rangle }F\big (\psi (t_n+\theta )\big )d\theta \Big ) \cdot \Big ( \int _0^{\sigma } e^{i(\sigma -\theta )\langle \partial _x\rangle } F\big (\psi (t_n+\theta )\big )d{\theta }\Big ) d{\xi }d{\sigma },\\ K_3&=-\int _0^{\tau }e^{i(\tau -\sigma )\langle \partial _x\rangle }i\epsilon ^4 {\langle \partial _x\rangle }^{-1}r_1\bigg ( e^{i\sigma \langle \partial _x\rangle }\psi (t_n)+\int _0^{\sigma } e^{i(\sigma -\theta )\langle \partial _x\rangle } \Big ({\mathscr {H}}\big (\psi (t_n+\theta )\big )\\&\quad +\epsilon ^4 i{\langle \partial _x \rangle }^{-1}r_1\big (\psi (t_n+\theta )\big ) +\dfrac{\epsilon ^4}{\beta ^4}i{\langle \partial _x \rangle }^{-1}r_2\big (\psi (t_n+\theta )\big )\Big )d{\theta }\bigg )d\sigma \\&\quad -\int _0^{\tau }e^{i(\tau -\sigma )\langle \partial _x\rangle }i\dfrac{\epsilon ^4}{\beta ^4}{\langle \partial _x\rangle }^{-1}r_2\bigg ( e^{i\sigma \langle \partial _x\rangle }\psi (t_n)+\int _0^{\sigma } e^{i(\sigma -\theta )\langle \partial _x\rangle }\Big ({\mathscr {H}}\big (\psi (t_n+\theta )\big )\\&\quad +\epsilon ^4 i{\langle \partial _x \rangle }^{-1}r_1\big (\psi (t_n+\theta )\big )+\dfrac{\epsilon ^4}{\beta ^4} i{\langle \partial _x \rangle }^{-1}r_2\big (\psi (t_n+\theta )\big )\Big )d{\theta }\bigg )d\sigma . \end{aligned}$$

We can reformulate \(K_1\) as

$$\begin{aligned} K_1&{=}i\tau \epsilon ^4 \int _0^{\tau }\partial _{\sigma } \Big (e^{i\sigma \langle \partial _x \rangle }{\langle \partial _x\rangle }^{{-}1}r_1\big (\psi (t_n)\big )\Big )d\sigma {+}i\tau \dfrac{\epsilon ^4}{\beta ^4}\int _0^{\tau }\partial _{\sigma } \Big (e^{i\sigma \langle \partial _x \rangle }{\langle \partial _x\rangle }^{{-}1}r_2\big (\psi (t_n)\big )\Big )d\sigma \\&=\tau \epsilon ^4 \int _0^{\tau }e^{i\sigma \langle \partial _x\rangle }r_1\big (\psi (t_n)\big )d\sigma +\tau \dfrac{\epsilon ^4}{\beta ^4} \int _0^{\tau }e^{i\sigma \langle \partial _x\rangle }r_2\big (\psi (t_n)\big )d\sigma , \end{aligned}$$

then we obtain that under the assumptions \(\mathrm (A)\) with \(m\ge 0\),

$$\Vert K_1\Vert _{1}\lesssim \tau ^2\alpha ^4.$$

where \(\alpha =\epsilon \) for \(\beta \ge 1\) and \(\alpha ={\epsilon }/{\beta }\) for \(\epsilon<\beta <1\). Similarly, by the definitions of the functions \(\mathscr {H}(\psi )\), \(r_1(\psi )\) and \(r_2(\psi )\), we can obtain estimates for \(K_2\) and \(K_3\) as \(\Vert K_2 \Vert _{1}\lesssim \tau ^2\alpha ^4\) and \(\Vert K_3 \Vert _{1}\lesssim \tau ^6\alpha ^4\), respectively. Combining the above results, we have

$$\Vert \mathscr {R}^n\Vert _{1}\lesssim \tau ^2\alpha ^4.$$

Obviously, (22) can be reformulated as

$$\begin{aligned} {\mathscr {Q}}\big (\psi (t_n)\big )&=\tau e^{i\tau \langle \partial _x \rangle } {\mathscr {H}}\big (\psi (t_n)\big ) - \int _0^{\tau }e^{i(\tau -\sigma )\langle \partial _x\rangle } {\mathscr {H}}\big ( e^{i\sigma \langle \partial _x\rangle }\psi (t_n)\big ) d\sigma \\&=\int _0^{\tau } \int _0^{\sigma } -\partial _{\theta }\Big ( e^{i(\tau -\theta ) \langle \partial _x \rangle }{\mathscr {H}}\big (e^{i\theta \langle \partial _x\rangle }\psi (t_n) \big )\Big )d\theta d\sigma . \end{aligned}$$

By the properties of the operator \(e^{i\sigma \langle \partial _x \rangle }\), we have

$$\begin{aligned} \Vert \mathscr {Q}(\psi (t_n))\Vert _{1}\lesssim \tau ^2\alpha ^2\Vert \psi (t_n)\Vert _{1}^2. \end{aligned}$$

From the above results, under the assumption \(\mathrm (A)\) we have that

$$\begin{aligned} \Vert \mathscr {Q}\big (\psi (t_n)\big )\Vert _{1} \lesssim \alpha ^2\tau ^2,\quad \Vert \mathscr {R}^n\Vert _{1} \lesssim \alpha ^4\tau ^2,\qquad 0\le n \le {T_\epsilon }/{\tau }. \end{aligned}$$
(2)

Then, we complete the proof of the local error bounds (23).

The proof of Theorem 3

Firstly we give the following lemma [25].

Lemma B.2

Given \(u,v \in X_M\) and \(\tilde{u}_l,\tilde{v}_l\) defined in (13), we have

$$\begin{aligned} h\sum _{j=0}^{M-1} \overline{u}_j v_j=(b-a)\sum _{l\in T_M}\overline{\tilde{u}}_l\tilde{v}_l. \end{aligned}$$
(1)

From Lemma B.2 and the energy expression (31), we get

$$\begin{aligned} \begin{aligned} E^n_h&=\sum _{l \in T_M} \frac{b-a}{2} \Big ( | (\widetilde{z^n})_l|^2+(3+\mu _l^2)|(\widetilde{u^n})_l|^2\Big )\\&\quad +\frac{h}{2}\sum _{j=0}^{M-1}\Big [\frac{2}{\epsilon ^2}\big (1-\cos (\epsilon u^n_j)\big )+\frac{\beta ^2}{\epsilon ^2}\big (1-\cos (\frac{2\epsilon }{\beta } u^n_j)\big )-3| u^n_j|^2\Big ]\\&= \dfrac{b-a}{2}\Big ( \Vert {z^n}\Vert _{0}^2+\Vert {u^n}\Vert _{1}^2 \Big )\\&\quad +\frac{h}{2}\sum _{j=0}^{M-1}\Big [\frac{2}{\epsilon ^2}\big (1-\cos (\epsilon u^n_j)\big )+\frac{\beta ^2}{\epsilon ^2}\big (1-\cos (\frac{2\epsilon }{\beta } u^n_j)\big )-3| u^n_j|^2\Big ], \end{aligned} \end{aligned}$$
(2)

then the discrete energy deviation from the initial energy is

$$\begin{aligned} \begin{aligned} |E^n_h-E^0_h|&= \frac{b-a}{2}\Big ( \Vert {z^n}\Vert _{0}^2- \Vert {z^0}\Vert _{0}^2+\Vert {u^n}\Vert _{1}^2 -\Vert {u^0}\Vert _{1}^2\Big )+\frac{h}{2}\sum _{j=0}^{M-1}\Big [\frac{2}{\epsilon ^2}\Big (1-\cos (\epsilon u^n_j)\\&\quad -\big (1-\cos (\epsilon u^0_j)\big )\Big )+\frac{\beta ^2}{\epsilon ^2}\Big (1-\cos (\frac{2\epsilon }{\beta } u^n_j)\\&\quad -\big (1-\cos (\frac{2\epsilon }{\beta } u^0_j)\big )\Big )-(3| u^n_j|^2-3| u^0_j|^2)\Big ]\\&:=E_1+E_2, \end{aligned} \end{aligned}$$
(3)

where

$$\begin{aligned} E_1&:=\frac{b-a}{2}\Big ( \Vert {z^n}\Vert _{0}^2- \Vert {z^0}\Vert _{0}^2+\Vert {u^n}\Vert _{1}^2 -\Vert {u^0}\Vert _{1}^2\Big )\\&=\frac{b-a}{2}\Big ( \Vert {z^n}-\zeta ^n+\zeta ^n\Vert _{0}^2- \Vert ({z^0}-\zeta ^0+\zeta ^0\Vert _{0}^2\\&\quad +\Vert {u^n}-U^n+U^n\Vert _{1}^2 -\Vert ({u^0}-U^0+U^0\Vert _{1}^2\Big )\\&\le \frac{b-a}{2}\Big ( \Vert {z^n}-\zeta ^n\Vert _{0}^2+\Vert \zeta ^n\Vert _{0}^2+2\sqrt{\Vert {z^n}-\zeta ^n\Vert _{0}^2\cdot \Vert \zeta ^n\Vert _{0}^2}- \Vert ({z^0}-\zeta ^0\Vert _{0}^2-\Vert \zeta ^0\Vert _{0}^2\\&\quad -2\sqrt{\Vert {z^0}-\zeta ^0\Vert _{0}^2\cdot \Vert \zeta ^0\Vert _{0}^2}+\Vert {u^n}-U^n\Vert _{1}^2+\Vert U^n\Vert _{1}^2+2\sqrt{\Vert {u^n}-U^n\Vert _{0}^2\cdot \Vert U^n\Vert _{0}^2} \\&\quad -\Vert {u^0}-U^0\Vert _{1}^2-\Vert U^0\Vert _{1}^2-2\sqrt{\Vert {u^0}-U^0\Vert _{0}^2\cdot \Vert U^0\Vert _{0}^2}\Big )\\&\le \frac{b-a}{2}\Big ( \Vert {z^n}-\zeta ^n\Vert _{0}^2+\Vert \zeta ^n\Vert _{0}^2-\Vert \zeta ^0\Vert _{0}^2+\Vert {u^n}-U^n\Vert _{1}^2+\Vert U^n\Vert _{1}^2-\Vert U^0\Vert _{1}^2\Big )\\&\quad +(b-a)(\Vert {z^n}-\zeta ^n\Vert _{0}+\Vert {u^n}-U^n\Vert _{0}) \end{aligned}$$

where \(\zeta ^n\) and \(U^n\) are the values of the exact solution \(\partial _t u(x,t)\) and u(xt) of DSGE(4) at \(t_n\), respectively, \(\Vert {z^0}-\zeta ^0\Vert _{0}^2=0\) and \(\Vert {u^0}-U^0\Vert _{1}^2=0\).

Applying inequality \(1-\cos (u)\le u^2/2\), we can get

$$\begin{aligned} E_2&:=\frac{h}{2}\sum _{j=0}^{M-1}\Big [\frac{2}{\epsilon ^2}\Big (1-\cos (\epsilon u^n_j)-\big (1-\cos (\epsilon u^0_j)\big )\Big )\\&\quad +\frac{\beta ^2}{\epsilon ^2}\Big (1-\cos (\frac{2\epsilon }{\beta } u^n_j)-\big (1-\cos (\frac{2\epsilon }{\beta } u^0_j)\big )\Big )-(3| u^n_j|^2-3| u^0_j|^2)\Big ]\\&=\frac{h}{2}\sum _{j=0}^{M-1}\Big [\frac{2}{\epsilon ^2}\Big (\big (1-\cos (\epsilon u^n_j)\big )-\big (1-\cos (\epsilon U^n_j)\big )+\big (1-\cos (\epsilon U^n_j)\big )\Big )\\&\quad -\frac{2}{\epsilon ^2}\Big (\big (1-\cos (\epsilon u^0_j)\big )-\big (1-\cos (\epsilon U^0_j)\big )+\big (1-\cos (\epsilon U^0_j)\big )\Big )\\&\quad +\frac{\beta ^2}{\epsilon ^2}\Big (\big (1-\cos (\frac{2\epsilon }{\beta } u^n_j)\big )-\big (1-\cos (\frac{2\epsilon }{\beta } U^n_j)\big )+\big (1-\cos (\frac{2\epsilon }{\beta } U^n_j)\big )\Big )\\&\quad -\frac{\beta ^2}{\epsilon ^2}\Big (\big (1-\cos (\frac{2\epsilon }{\beta } u^0_j)\big )-\big (1-\cos (\frac{2\epsilon }{\beta } U^0_j)\big )+\big (1-\cos (\frac{2\epsilon }{\beta } U^0_j)\big )\Big )\\&\quad -(3| u^n_j|^2-3| u^0_j|^2)\Big ]\\&\le \frac{h}{2}\sum _{j=0}^{M-1}\Big [ \frac{2}{\epsilon ^2}(\dfrac{\epsilon ^2|u^n_j|^2}{2}-\dfrac{\epsilon ^2|U^n_j|^2}{2})-\frac{2}{\epsilon ^2}( \dfrac{\epsilon ^2|u^0_j|^2}{2}-\dfrac{\epsilon ^2|U^0_j|^2}{2})\\&+\dfrac{2}{\epsilon ^2}\big ((1-\cos (\epsilon U^n_j))-(1-\cos (\epsilon U^0_j))\big )+\frac{\beta ^2}{\epsilon ^2}(\dfrac{4\epsilon ^2|u^n_j|^2}{2\beta ^2}-\dfrac{4\epsilon ^2|U^n_j|^2}{2\beta ^2})\\&\quad -\frac{\beta ^2}{\epsilon ^2}(\dfrac{4\epsilon ^2|u^0_j|^2}{2\beta ^2}-\dfrac{4\epsilon ^2|U^0_j|^2}{2\beta ^2}) +\frac{\beta ^2}{\epsilon ^2}\big ((1-\cos (\frac{2\epsilon }{\beta } U^n_j))+(1-\cos (\frac{2\epsilon }{\beta } U^0_j))\big )\\&\quad -(3| u^n_j|^2-3| u^0_j|^2)\Big ]\\&=\frac{h}{2}\sum _{j=0}^{M-1}\Big [ -(3|U^n_j|^2-3|u^0_j|^2)+\dfrac{2}{\epsilon ^2}\big ((1-\cos (\epsilon U^n_j))-(1-\cos (\epsilon U^0_j))\big )\\&\quad +\frac{\beta ^2}{\epsilon ^2}\big ((1-\cos (\frac{2\epsilon }{\beta } U^n_j))+(1-\cos (\frac{2\epsilon }{\beta } U^0_j))\big )\Big ]. \end{aligned}$$

Under the assumption (A), combining the estimates of \(E_1\) and \(E_2\), we can obtain

$$\begin{aligned} |E^n_h-E^0_h|&\le \frac{b-a}{2}\Big ( \Vert {z^n}-\zeta ^n\Vert _{0}^2+\Vert \zeta ^n\Vert _{0}^2-\Vert \zeta ^0\Vert _{0}^2+\Vert {u^n}-U^n\Vert _{1}^2\\&\quad +\Vert U^n\Vert _{1}^2-\Vert U^0\Vert _{1}^2\Big )+(b-a)(\Vert {z^n}-\zeta ^n\Vert _{0}+\Vert {u^n}-U^n\Vert _{1 })\\&\quad +\frac{h}{2}\sum _{j=0}^{M-1}\Big [ \dfrac{2}{\epsilon ^2}\big ((1-\cos (\epsilon U^n_j))-(1-\cos (\epsilon U^0_j))\big )\\&\quad +\frac{\beta ^2}{\epsilon ^2}\big ((1-\cos (\frac{2\epsilon }{\beta } U^n_j))+(1-\cos (\frac{2\epsilon }{\beta } U^0_j))\big )-(3|U^n_j|^2-3|u^0_j|^2)\Big ], \end{aligned}$$

since the exact solution of the DSGE (4) satisfies conservation of energy, therefore

$$\begin{aligned} |E^n_h-E^0_h|{\le } \frac{b-a}{2}\Big ( \Vert {z^n}{-}\zeta ^n\Vert _{0}^2{+}\Vert {u^n}-U^n\Vert _{1}^2\Big ){+}(b-a)(\Vert {z^n}-\zeta ^n\Vert _{0}+\Vert {u^n}-U^n\Vert _{1}),\nonumber \\ \end{aligned}$$
(4)

according to the error estimation results (1), we have the following estimates for the discrete energy:

$$\begin{aligned} |E^n_h-E^0_h|\lesssim h^m+\alpha ^2\tau +\tau _0^{m+1},\quad 0\le n\le T_\epsilon /\tau . \end{aligned}$$
(5)

In addition, if the exact solution is sufficiently smooth, e.g., \(u,\partial _t u \in H^{\infty }\), the estimate for the discrete energy deviation for sufficiently small \(\tau \) is

$$\begin{aligned} |E^n_h-E^0_h|\lesssim h^m+\alpha ^2\tau ,\quad 0\le n\le T_\epsilon /\tau . \end{aligned}$$
(6)

The proof of (32) is completed.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, L., Song, H. & Yi, W. Improved Uniform Error Bounds on a Lawson-type Exponential Integrator Method for Long-Time Dynamics of the Nonlinear Double Sine-Gordon Equation. J Sci Comput 102, 25 (2025). https://doi.org/10.1007/s10915-024-02752-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-024-02752-6

Keywords

Mathematics Subject Classification